# Difference between revisions of "M(8,4,3)"

M(8,4,3) - $kSL_2(3)$
Representative: $kSL_2(3)$ $Q_8$ $C_3$ 7 3 1 $\left( \begin{array}{ccc} 4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 4 \\ \end{array} \right)$ Yes Yes Yes $\mathcal{O}SL_2(3)$ $\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \\ \end{array}\right)$ 1 $\mathcal{T}(B)=S_3$[1] {{{PIgroup}}} No Yes M(8,4,2) No

These are tame blocks, and appear in the family $Q(3 {\cal K})$ in Erdmann's classification (see [Er88a], [Er88b]). The class lifts to a unique $\mathcal{O}$-Morita equivalence class by [HKL07]. A derived equivalence with M(8,4,2) over $k$ was established in [Ho97].

## Basic algebra

Quiver: a:<1,2>, b:<2,3>, c:<3,1>, d:<2,1>, e:<3,2>, f:<1,3>

Relations w.r.t. $k$: ab=fcf, bc=dad, ca=ebe, fe=ada, df=beb, ed=cfc, dab=0=bed=cfe

## Irreducible characters

$k_0(B)=4, k_1(B)=3$

## Notes

1. ${\rm Pic}(B)=\mathcal{L}(B)$ by [LiMa20b], so ${\rm Pic}(B)=\mathcal{T}(B)$